String topology in dimensions two and three

Let V denote the vector space with basis the conjugacy classes in the fundamental 4 group of an oriented surface S. In 1986 Goldman [1] constructed a Lie bracket [,] on V. If a and b are conjugacy classes, the bracket [a; b] is defined as the signed sum over intersection points of the conjugacy classes represented by the loop products taken at the intersection points. In 1998 the authors constructed a bracket on higher dimensional manifolds which is part of String Topology [2]. This happened by accident while working on a problem posed by Turaev [3], which was not solved at the time. The problem consisted in characterizing algebraically which conjugacy classes on the surface S are represented by simple closed curves. Turaev was motivated by a theorem of Jaco and Stallings [4,5] that gave a group theoretical statement equivalent to the three dimensional Poincaré conjecture. This statement involved simple conjugacy classes. Recently a number of results have been achieved which illuminate the area around Turaev's problem. Now that the conjecture of Poincare has been solved, the statement about groups of Jaco and Stallings is true and one may hope to find a Group Theory proof. Perhaps the results to be described here could play a role in such a proof. See Sect. 3 for some first steps in this direction.